During operation, many systems and their constituent components are subjected to both mechanical and thermal stresses. Individual components, for example, can be subjected to direct mechanical stresses through the application of compressive or tensile forces. Thermal stresses occur due to temperature changes directly within the component or in the environment in which the component operates. Such stresses may be constant or vary as a function of time. The components in a gas turbine are subjected to such cyclic mechanical and thermal stresses, especially when the gas turbine is started or shut down.
Extreme cyclic loading, both mechanical and thermal, results in material fatigue, which, in many cases limits the life of one or more components exposed to the loading. Fatigue crack growth (FCG) under cyclic loading conditions is often the life-limiting mechanism for a component. Small cracks can nucleate from inherent material flaws, such as a preexisting flaw in a forging or from another crack-initiation mechanism, such as low-cycle fatigue (LCF).
Cyclic loading, for example the starting and shutting down of an engine or another significant change in its operating condition, may cause small cracks in a component to grow incrementally without directly and immediately impacting the structural integrity of the component. This phenomenon is referred to as stable crack growth. But when the crack reaches a critical size, its growth becomes unstable (i.e., growth in an uncontrolled manner). The unstable crack grows quickly and significantly, resulting in possible component failure.
The number of cycles N at which unstable crack growth begins is called the fatigue crack life of the component. The crack growth rate, which affects the value of N, can be estimated using linear elastic fracture mechanics (LEFM) and finite element analyses (FEA) for estimating a transient stress field to which the component is subjected.
Due to the uncertainties associated with material properties and initial flaw size and the complexities of the LEFM and FEA analyses, estimating a lifetime of a component is a difficult and tedious process. Therefore, the design of some components may not consider fatigue crack growth or the design may be extremely conservative relative to fatigue crack growth. As a result of this conservative approach, the component may be designed with conservative features (material, dimensions, tolerances, etc.), its operating conditions may be limited (minimum starting metal temperatures, etc.), or it may be prematurely serviced or taken out of service, i.e., when a more accurate analysis may permit a longer service interval. Such premature service intervals or component replacements may add significantly to system cost.
Fatigue crack life calculations can be made according to a deterministic or a probabilistic approach.
The deterministic approach uses minimum (or maximum, as appropriate) material properties to estimate a component service life. Conservative estimates of material properties and initial flaw sizes are used, as well as worst-case scenarios and significant safety factors. For instance, a gas turbine includes, according to one design, about twenty rotor disks (also referred to as compressor disks or turbine disks) stacked horizontally end-to-end to form a gas turbine rotor. See FIG. 1 and the discussion of FIG. 1 below. The deterministic approach may yield a fatigue crack life of N=3000 engine starts. This result is based on minimum material properties and maximum assumed flaw size at the worst possible location (largest stress range) on the component, e.g., the rotor disk. Using worst case assumptions, the fatigue crack life N can be conservatively estimated by LEFM or known extensions of that technique. This approach falls under the so-called safe-life design philosophy and has been used for land-based heavy duty gas and steam turbines.
The drawback of such deterministic fracture mechanics calculations is that the analysis of a component is based on a single location or a few locations and on minimum/maximum material properties at those locations. The distribution of the material properties and flaw sizes throughout the component is not used in the deterministic approach. The safety-factor, which is inherent in the deterministic approach, may thereby lead to an overly conservative design.
The second approach to determining component life involves probabilistic methods and statistics to study the influence of thermal and mechanical stress variations on a component. In particular, a probabilistic analysis may utilize fracture mechanics calculations based on flaw-size distributions and component inspection intervals. The probabilistic approach may permit the design of lower cost components with expanded operating conditions and/or longer service lifetime than permitted by the deterministic approach.
Advantages and disadvantages of the probabilistic and deterministic approaches have been discussed at length in the pertinent literature. Both approaches can be used to conduct failure analyses of a gas or combustion turbine and its constituent components, in particular its rotating turbine disks. A gas or combustion turbine is a type of internal combustion engine. An air stream is compressed and accelerated within a compressor. Fuel is injected into the air stream in a combustor or combustion chamber where ignition of the fuel occurs. Ignition of the fuel creates a hot combustion gas flow that is directed to a turbine and causes it to rotate. The combustion gas stream (also referred to as a working gas) expands as it enters the turbine, which includes rows of stationary guide vanes and rotating blades connected to a turbine shaft. The expanding gas flow is accelerated by the guide vanes and directed over the rotating blades, causing the blades and thus the turbine shaft to spin. The spinning shaft turns the compressor and also provides a mechanical torque output. After passing through the turbine disks, the working gas flow enters a turbine exhaust casing.
FIG. 1 depicts a prior art gas or combustion turbine 10, generally including a compressor 12, a combustion chamber 14 and a turbine 16. The compressor 12 inducts and compresses ambient air. The compressed air then enters one or more combustors 28 in the combustion chamber 14, where the compressed air is mixed with fuel. The air-fuel mixture ignites to form a hot working gas. The working gas is directed to the turbine 16 where it expands through alternating rows of stationary guide vanes 22 and rotating blades 18 to generate mechanical forces that turn a shaft, which is not specifically shown in FIG. 1. The expanded gas exits the turbine 16 via an exhaust casing (not shown). The rotating blades 18 are attached to rotor disks 40 that are in turn affixed to the turbine shaft.
In a probabilistic analysis of gas turbine rotor disks, variations in material properties, flaw size and location distribution are used to determine a probability of failure, PoF(N), after N operational cycles. A typical probability-of-failure value for a gas turbine rotor disk after N=3000 starts is on the order of:PoF(3000)˜1/1,000,000This result indicates that after about 3000 starts, 1 of 1,000,000 rotor disks will have failed.
Another important metric for component life is the hazard H(N) or the risk per start, which for low PoF(N) values is a derivative dPoF(N)/dN.
Other types of probabilistic analyses, such as probabilistic low cycle fatigue (LCF) analysis, are of interest as well.
To date, a few publicly known probabilistic tools have been developed for use in only two-dimensional analyses as probabilistic analysis is still in its infancy. For example, the well-known DARWIN code can be used only for 2-dimensional axially-symmetric components and is geared towards aero (e.g., aeronautic and aerospace) engine design. This code employs a zone based approach that spatially decomposes the components into a number of zones. A theoretical crack is positioned within each zone for representing the crack growth for the entire component.
In contrast to the DARWIN code, the ProbFM code, which is described in a commonly-owned patent application entitled Method and System for Probabilistic Fatigue Crack Life Estimation (Publication No. US 2014/0107948 A1, application Ser. No. 13/652,671) is a direct simulation Monte-Carlo approach that uses high-performance computing techniques to perform billions of fracture mechanics simulations, positioning cracks everywhere in the component. The corresponding crack growth path is assumed according to the calculated stress field (i.e. according to the first principal stress field). No manual positioning of cracks or other manual processes are needed. The evaluations can be replicated easily by different design engineers using the ProbFM code. Publication No. US 2014/0107948 A1 is incorporated by reference herein in its entirety. To date both codes DARWIN and ProbFM work properly only in two-dimensions.
In the area of probabilistic fracture mechanics, the extension from two-dimensional techniques to three-dimensional techniques is very involved and no techniques are known. A gas turbine rotor disk can be analyzed by the aforementioned DARWIN program only by analyzing the axis-symmetric two-dimensional problems, or by the ProbFM code, but again only in two dimensions.